Ultra-precision robotic system

ABSTRACT

In this invention, an ultra-precision robotic system yielding either rigid body guidance or large deformation analysis (LDRS, i.e. semi-flexible and flexible robotics) is developed based on the technology of shadow robotic measurement system (shadow system) and robotic real-time stepping convergence and control. With an extreme case of an uncorrelated 6-D revolute joint shadow system, it indicates that the positioning accuracy of such an ultra-precision robotic system will not be in excess of {7.21 nm, 1.68 arc-second}. With the general case of a Puma 560 shadow system, the positioning accuracy also appears not in excess of {7.21 nm, 1.68 arc-second} in most of its working space. By more conservative estimate in practice, the error of positioning accuracy, ERRε[( 50 nm, 2 arc-second), (10 −1  μm, 3 arc-second)] are expected with current manufacturing and logistical conditions.

BACKGROUND OF THE INVENTION

[0001] This invention is related to a robotic system with rigid body guidance or a robotic system with large deformation analysis (LDRS, i.e. semi-flexible and flexible robotic system) in robotics. A robotic system possesses the advanced properties to be one of the most important equipments in modem precision engineering. However, there are some serious technical barriers which have greatly dragged down the qualification of positioning accuracy of the technology for its general applications in precision engineering. The following degrading influences are considered as the major problems to cause such barriers:

[0002] The influence of tolerance and clearance;

[0003] The influence of the deformation of elements under load;

[0004] The influence of the change of load;

[0005] The influence of the wear of kinematic pair;

[0006] The influence of the ambient conditions, especially, the temperature; and

[0007] The influence of some associated problems with the gap of clearance and wear (i.e. backlash, contamination, and the thin film of lubrication, etc.).

[0008] If these problems can be resolved, it will be easy for a robotic system to achieve the precise positioning accuracy or even ultra-precision positioning accuracy, and to claim its important role in modern precision engineering. The difficulty is that these degrading factors are inevitable with current understanding of the technology of robotics since they are associated with the nature of design, manufacturing, assembly, and application of a robotic system. As a result, to date, the positioning accuracy of a robot is relatively low. For example, in the development of a precise robotic computed tomography inspection system, a robotic system with the positioning accuracy in two-digital micrometer-level {0.001 in (0.025 mm), 5 arc-second} is urgently needed; however, on today's robotic market, such a precise robot is still unavailable.

[0009] If a robotic system yielding rigid body analysis is not qualified to have precision positioning accuracy, then a robotic system yielding large deformation analysis (LDRS, i.e. the semi-flexible and the flexible robotic systems) is almost impossible to play a role in precision engineering. With a LDRS, all the above degrading problems exist; moreover, the problem of deformation becomes extremely serious. The uncertainties of nonlinear correlations of deformation make the LDRS too difficult to be controlled. By now, the technology of LDRS is still at its very early developing stage. Although a LDRS can have higher payload/weight ratio and better dynamic properties, which are considered as the future direction of development of modern robotics, it is noted that no 6-D flexible robotic system has been developed yet in real application with the acceptable accuracy.

[0010] With investigation, it is found that many valuable efforts of robotic system control with academic researches and industrial practices have been done for the development of both the rigid body robotic system and the LDRS. For simplification, the prior efforts can be classified into two different categories of control technology.

[0011] The first is the category of direct control theory. For example, the robotic control method under rigid body guidance, the robotic control method with finite element analysis, the robotic control method based on real-time monitoring with embodied sensor on the elements of kinematic chain are the major efforts with this category. The direct control technology focuses on the understanding of the properties and characteristics of the researched robotic system itself to explore the effective control. As above-mentioned, a robotic system with direct control suffer the influences of degrading factors, and it is difficult for it to achieve ultra-precision positioning accuracy.

[0012] The other can be classified as the category of indirect control theory. This technology tries to develop the effective control with the help of some accessories that are not necessarily defined as part of the studied robotic system. The oldest indirect control of the robotic system is the tactile sensor system, which uses contact sensors to detect the position of the robotic system in work cells and then to form feedback control loop for robotic system control. Currently, the most popular indirect control theory may be the visual servoing technology for robotic system control. Since this technology can form a non-contact feedback closed-loop, it is getting its dominating position in the development of robotic system control. Generally, the visual servoing technology for robotic system control can be simplified in three different ways. The first is the on-body method. With this method, the visual sensor is mounted on the body of the robotic system to gather feedback control data. The second is the on-work-cell method. With this method, the visual sensor is mounted somewhere in the work cell to monitor the robotic system for gathering feedback-control data. The third is the combination of the above two ways.

[0013] Currently, visual servoing technology of robotics is considered the main stream of robotic motion control. However, so far, visual servoing technology used in robotics can only control relatively simple objects undergoing constrained motion, or simple motion for complex objects. A majority of the visual servoing systems continue to employ artificial markers to circumvent the end-effecter and positioning data. The recent advances have allowed the development of theoretical frameworks for more complicated problems, the demonstration of real-time control for relative complex applications, and the construction of servoing systems that use no or very limited markers for the tracking process. However, the visual technology is still in developing process, and it is difficult for the technology to achieve a positioning accuracy in the micro-precision level at this stage.

[0014] From the above discussions, the existing technologies in direct or indirect robotic control appear difficult to achieve ultra-precision positioning accuracy. In such a situation, it calls for more ideas and innovative researches. It is therefore an object of this invention to develop a robotic system to have the capability to achieve ultra-precision positioning accuracy to overcome all the influences of the degrading factors to claim the important role of robotic system in modern precision engineering.

BRIEF SUMMARY OF THE INVENTION

[0015] In this invention, an ultra-precision robotic system is developed based on the technology of shadow robotic measurement system [1]. Following the above classification, it can be classified as in the indirect robotic control technology. It is characterized by providing some significant features of development with the application of the advantages of shadow robotic measurement system. In general, the invention is to provide the precision engineering an advanced and useful device.

[0016] One of the features of the invention is to provide an apparatus to reduce the influence of tolerance and clearance associated with the nature of a traditional robot in an insignificant level to meet the requirements of the development of ultra-precision robotic system

[0017] Still one of the features of the invention is to provide an apparatus to eliminate the influence of the deformation of elements under load associated with the nature of a traditional robot in an insignificant level to meet the requirements of the development of ultra-precision robotic system

[0018] The other feature of the invention is to provide an apparatus to eliminate the influence of the change of load associated with the nature of a traditional robot in an insignificant level to meet the requirements of the development of ultra-precision robotic system

[0019] Still the other feature of the invention is to provide an apparatus to reduce the influence of the wear of kinematic pair associated with the nature of a traditional robot in an insignificant level to meet the requirements of the development of ultra-precision robotic system

[0020] Another feature of the invention is to provide an apparatus for the convenient applications of the technical means to reduce the influence of the ambient conditions, especially, the temperature associated with the nature of a traditional robot in an insignificant level to meet the requirements of the development of ultra-precision robotic system

[0021] Still another feature of the invention is to provide an apparatus to reduce the influence of some associated problems with the gap of clearance and wear (i.e. backlash, contamination, and the thin film of lubrication, etc.) associated with the nature of a traditional robot in an insignificant level to meet the requirements of the development of ultra-precision robotic system

[0022] The invention with its organization, working method, manner of operation, and utilization can be best understood by making reference to the following description of the drawings and the depictions of the invention.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0023]FIG. 1 is a general 3-D view of an embodiment of the invention.

[0024]FIG. 2 is an overall drawing of an embodiment of the invention with the top view, side view, front view, and rear view.

[0025]FIG. 3 is the drawing of an extremely built 6-D rotational joint robot for error analysis.

[0026]FIG. 4 is the transformation of the link frames of robotic system and shadow robotic measurement system.

[0027]FIG. 5 is the drawing of a Puma 560 Type Shadow Robotic Measurement System for general for error analysis.

[0028]FIG. 6 is the computer simulation result of analytical errors with the joint configuration of Θ={θ₁=0˜1080°; θ₂=218°; θ₃=210°; θ₄=71°; θ₅=84°; θ₆=39°}.

[0029]FIG. 7 is the computer simulation result of analytical errors with the joint configuration of Θ={θ₁=67°; θ₂=0˜1080°; θ₃=214°; θ₄=172°; θ₅=183°; θ₆=40°}.

[0030]FIG. 8 is the computer simulation result of analytical errors with the joint configuration of Θ={θ₁=88°; θ₂=18°; θ₃=0˜1080°; θ₄=174°; θ₅=181°; θ₆=132°}.

[0031]FIG. 9 is the computer simulation result of analytical errors with the joint configuration of Θ={θ₁=186°; θ₂=118°; θ₃=22°; θ₄=0˜1080°; θ₅=12°; θ₆=92°}.

[0032]FIG. 10 is the computer simulation result of analytical errors with the joint configuration of Θ={θ₁=86°; θ₂=318°; θ₃=92°; θ₄=73°; θ₅=0˜1080°; θ₆=12°}.

[0033]FIG. 11 is the computer simulation result of analytical errors with the joint configuration of Θ={θ₁=217°; θ₂=38°; θ₃=193°; θ₄=91°; θ₅=232°; θ₆=0˜1080°}.

DETAILED DESCRIPTION OF THE INVENTION

[0034] Shadow Robotic Measurement System of the Invention

[0035] In this invention, an ultra-precision robotic system is developed based on the development of the novel technology of shadow robotic measurement system (shadow system), which can effectively resolve the above-mentioned unavoidable degrading problems to achieve the ultra-precision positioning accuracy with no significant difficulties.

[0036]FIG. 1 depicts that such an ultra-precision robotic system. In FIG. 1, (1) is an embodiment of shadow robotic measurement system, (2) is an embodiment of robot, and (3) is the robotic control with corresponding computer system.

[0037]FIG. 2 depicts the basic structure of such an ultra-precision robotic system with more detailed different views. In FIG. 2, it shows that for a 6-DOF robot, a shadow system with at least 6 DOF is built to match the said 6-D robot. The said 6-D robot and the shadow system are connected together through their original positions of the end-effecter frame. The shadow system is a passive system. Under the pulling force of the robot acting on connection-point (CP) at the origin of the end-effecter frame, the shadow system can freely follow the motion of the robot and stand no load.

[0038] With this embodiment in FIG. 2, all the joints of the shadow system are revolute, and each of the revolute joints is mated with a corresponding rotary measurement sensor to record the relative angular displacements of the joint. For achieve ultra-precision measurement, here, the optical rotary encoder is recommended to detect the angular displacements. With the angular displacements of the joints detected by the rotary encoders, the shadow system can then independently determine the position of end-effecter frame of the matched robot. Generally, a shadow robotic measurement system may have both revolute and prismatic joints. In such a case, the revolute joint should is mated with a corresponding rotary measurement sensor to record the angular displacement, and the prismatic joint should is mated with a corresponding linear measurement sensor to record the linear displacement.

[0039] From FIG. 2, it is known that the shadow system has formed a control loop independent of the kinematic chain of the robot to determine the position of the end-effecter of robot. It means that all these degrading factors associated with the mated robot have nothing to do with the control loop formed by the shadow system. The control loop of the shadow system can communicate with the matched robot through the connected position of the end-effecter frame. In this way, the degrading factors of the robot can be treated with a black box effect, and only the combined influence of final position change of the end-effecter frame is needed as the feedback signal of the closed-loop control. With the feedback, the robotic control system can determine the actual position of the end-effecter frame of the robot; therefore, it can then excite its actuators step by step to finally drive the end-effecter frame to the designated position. This control process and algorithm is defined as a position-targeted real-time closed-loop feedback control associated with the shadow robotic measurement system. The working process of the shadow system is further described in detail as follows:

[0040] When the 6-D robotic system in FIG. 1 and FIG. 2 is needed to send the end-effecter of the robot to a desired position, at first, it finds the inverse kinematics solutions of the robot and excites its actuators yielding the solutions of inverse kinematics to move the end-effecter frame to that desired position. If the robotic system is ideal, then the end-effecter can exactly reach the desired position without error. However, because of these above-mentioned inevitable degrading problems, the end-effecter frame is not able to achieve the desired position yielding the ideal solutions of inverse kinematics, and the robotic system is not even able to know the position of the end-effecter frame of the robot in the working process. In this case, since the shadow robotic measurement system is ideal, it can detect the position of the end-effecter. With the detected data by the shadow system, the robotic control system is able to determine the differences between the actual position of the end-effecter of the robot and the desired position. By using the differences as feedback-control data, the robotic control system can then move the end-effecter of robot frame to reach the designated position. With continuation of the process step by step, the robotic system can finally achieve the desired position.

[0041] It is clear that the positioning accuracy of the robotic system totally relies on the measurement accuracy of the shadow system. The following discussions is for the better understanding of why the shadow system can be considered ideal, and why it can achieve the ultra-precision measurement accuracy to meet the requirements for the development of an advanced ultra-precision robotic system:

[0042] No Significant Deformation Under Loads:

[0043] It is noted that the shadow robotic measurement system passively follows the motion of the matched robot, and the change of the loads acting on the robot has nothing to do with the shadow system. Since it stands no working loads, the shadow system can be made very light, and all the rotary encoders, joints, and the links in the shadow system will stand only insignificant frictional forces, gravity, and dynamic forces caused by the light mass of itself. It means the deformation with the system can be ignored because the linkages of the system are relatively strong as compared with these insignificant forces. In this way, the shadow system can then be considered an ideal rigid system.

[0044] The Influence of Tolerance Error can be Controlled in NM Level:

[0045] There are two kinds of tolerances with the shadow robotic measurement system. One is the manufacturing tolerance and the other is the assembly tolerance.

[0046] The manufacturing tolerance is designed for the dimensional restriction of manufacturing a component. Once the element is manufactured, its dimension is unchangeable yielding the range of tolerance. After manufacturing, the actual dimension of tolerance can be measured and merged with the old structural nominal parameter as the new nominal parameter. The assembly tolerance is designed for the dimensional restriction of assembling related components. Also, once the components are assembled, the dimensions to mark the positions of the fixed components are unchangeable yielding the range of tolerance. After assembly, the actual dimension of tolerance can be measured and merged with the old structural nominal parameter to form the new nominal parameter.

[0047] The error to measure the new nominal parameter is the accuracy of the measurement instrument. It is known that the error of dimensional measurement can be controlled in 1 nm level or more precise if necessary with a specialized laser interferometer measurement platform.

[0048] The Influence of Clearance Error can be Controlled in Ultra-Precision Level:

[0049] Two kinds of clearance exist in the shadow robotic measurement system. One is the assembly clearance, and the other is the motion clearance for kinematic pairs.

[0050] The assembly clearance is designed for easily assembling components together. After the assembly, the components are fixed and there is no relative motion between the components. As in the previous case, once the components are fixed, the relative dimensions between components are unchangeable. The actual dimension of assembly clearance corresponding to fixed components can be measured and merged with the old structural nominal parameter as the new nominal parameter. The error of new dimensional parameter is the accuracy of the measurement instrument, which can be controlled in 1 nm level or more precise if necessary.

[0051] The motion clearance for kinematic pair is designed for guaranteeing the relative motion between connected components. After the assembly, there will be uncertain gap for the relative motion between the components. The gap is the major problem to degrade the positioning accuracy, and is also the source for contamination, backlash, and thin film of lubrication. Usually, this problem is unavoidable. However, it is noted that, if there is no force or only a very small force acting on the motion pair, a light duty ultra-precision bearing sets can be used to form a revolute joint pair. In such a case, the rollers can rotate on races with pure rolling of non-gap motion, and the error of clearance between the rolling rollers and the races are insignificant and can be ignored.

[0052] Measured Joint Rotational Displacement Error:

[0053] Since an ultra-precision rotary optical encoder can be mated with the ultra-precision bearing mounted at the revolute joint pair, it means that the measurement error of the rotational displacement yields the error of the ultra-precision optical encoders. On the current market, the resolution of the existing optical rotary encoder with 3.5 in diameter can achieve 4,608,000 counts/revolution, that is 0.28 arc-second. Since the encoder is directly mounted on the joint to measure the relative angular displacement of joint shaft and joint house, the measurement accuracy is the resolution of the encoder.

[0054] Rough Estimation of the Measurement Accuracy of the Shadow System:

[0055] The measurement accuracy of a shadow robotic measurement system can be roughly estimated. If a 6-D robotic system with 6 dimensional linkages is adopted to build the passive shadow system, the measured maximum linear error will not be larger than ${\Delta \quad D_{x}} = {\pm {\sum\limits_{i = 1}^{6}\quad {\Delta_{1}}}}$

[0056] with an extreme design of robotic system in which the 6 links are arranged in a line (FIG. 3). Here, ΔD_(X) is the linear dimensional error in X axis direction, and Δ_(i) is the measurement error to measure the element i. If necessary, Δ_(i) can be controlled in 1 nm level or better. The total error is 6 nm.

[0057] With FIG. 3, the linear dimensional error in Y direction is caused by the rotational error, and the measured maximum linear dimensional error is ${\Delta \quad D_{Y}} = {\pm {\sum\limits_{i = 1}^{6}\quad {{L_{1}{\Delta\theta}_{1}}}}}$

[0058] If the above optical rotary encoder is used, and each of the links of the robot is 500 mm, then the maximum linear dimensional error caused by the rotational error is 4 nm. The error analysis of the two extreme cases error will be combined to determine the maximum linear dimensional error.

[0059] Also, if a 6-D robotic system with 6 rotational joints is adopted to build the passive shadow robotic measurement system, the measured maximum rotational error of pitch without the influence of structural correlation will not be in excess of ${\Delta \quad \theta} = {\pm {\sum\limits_{i = 1}^{6}\quad {{\Delta\theta}_{1}}}}$

[0060] with the extreme design of the robotic system that all the joints rotate around Y axis in the same direction or reversed direction (FIG. 3). Here, Δθ is the rotational displacement measurement error of pitch, and Δθ_(i) is the measurement error to measure the rotational displacement of joint i. Δθ_(i) yields the measurement accuracy of 0.28 arc-second if the above-mentioned optical rotary encoder existing on the current market is used to build the monitoring system of the joint angular displacement.

[0061] The estimate shows that the major errors of the shadow system caused by the measurement of rotational error and the tolerance and clearance with the extreme design of robot are

ERR={ΔD, Δθ}=±{7.21 nm, 1.68 arc-second}

[0062] Since the rough estimate is based on the extreme case of 6-D robotic system with 6 rotational joints without the influence of structural correlation, it is noted that the structural correlation with a real robotic system may worsen the error. However, it is also noted that a real robotic system will not be built with all links in a line and all joints around one direction of a rotational axis as the one in FIG. 3 to cause the extreme error. By taking the average of the negative and positive factors, it is estimated that the total errors with a non-extreme case of real robotic system should be around the same level of the above-discussed extreme case. Based on the above discussions, at more conservative estimate, the following two goals for a 6-D robotic system are expected:

[0063] a. High expectation: ERR={0.00005 mm (50 nm), 2 arc-second}

[0064] b. Low expectation: ERR={0.0001 mm (10⁻¹ μm), 3 arc-second}

[0065] The estimate is based on the existing optical rotary encoder on the common market with the current manufacturing and logistical conditions. If a specialized ultra-precision rotary encoder can be found, the positioning accuracy with the technology in this research will be higher.

[0066] No Wear between Kinematic Joints:

[0067] Since there is no load on the shadow system and the frictional forces acting on the joints are insignificant, the wear of the kinematic joints can be ignored for quite a long time period.

[0068] The Shadow System can have a Good and Stable Working Condition:

[0069] Since the shadow robotic measurement system is independent, it is easy to provide an isolated environment for it to have good and stable working conditions. In this case, the work cell environment and the ambient conditions, especially the temperature, will not significantly drag down the positioning accuracy of the robotic system. Generally speaking, the temperature error may degrade the positioning accuracy. To guarantee the success of the project and prevent the system from the influence of ambient conditions, the following four methods basically used in current engineering practices will be considered.

[0070] 1. Use cooling agent to control the change of temperature caused by heat sources

[0071] 2. Restrict the working cell of the system in the designated working condition

[0072] 3. Have an isolation design to isolate the system from all the heat sources

[0073] 4. Design an ambient condition control system to cover the equipment

[0074] Generally speaking, these methods are effective to control the temperature error for the robotic system in the designated ambient conditions to achieve ultra-precision accuracy. It is noted that with a robotic system the influence of temperature is relatively unimportant since there is no significant heat sources involved in itself.

[0075] Inverse Kinematics of the Invention

[0076] With reference of FIG. 2, the inverse kinematics of the 6-D robotic system to move its end-effecter frame to a desired position with the feedback closed-loop control of the 6-D shadow robotic measurement system can be demonstrated as follows (FIG. 4).

[0077] In FIG. 4, {T_(fR0)} is the zero position of the robotic system; {T_(S0)} is the zero position of the shadow robotic measurement system; {T_(D)} is the desired position; {T_(T)} is the end-effecter frame position; {T_(fRi)}is the joint frame position of robotic system, and i=1, 2, . . . , 6; and {T_(sj)} is the joint frame position of the shadow system, and j=1, 2, . . . , 6.

[0078] In FIG. 4, the transformation from {TfR0} to {T_(S0)} is  _(S0)^(R0)T,

[0079] the transformation from {T_(S6)} to {T_(T)} is _(T) ⁶T_(S), and the transformation from {T_(fR6)} to {T_(T)} is _(T) ⁶T_(R). These transformation forms are determined in association with the design and manufacturing and all are known. The link transformation of the robot can be expressed as

₆ ⁰ T _(fR)=₁ ⁰ T _(fR) ¹ ₂ ¹ T ₃ ² T _(fR 4) ³ T _(fR 5) ⁴ T _(fR 6) ⁵ T _(fR)

_(T) ⁰ T _(fR)=₆ ⁰ T _(fR T) ⁶ T _(fR)

[0080] Because of the errors caused by the influences of degrading factors, ₆ ⁰T_(fR) it is unknown, and so is _(T) ⁰T_(fR).

[0081] To resolve the inverse kinematics, a virtual robot is imaged to implement the same positioning process. The virtual robot is the same as the real robotic system; however, it is an ideal robot with no error. The transformation of the ideal virtual robot is

₆ ⁰ T _(R)=₁ ⁰ T _(R 2) ¹ T _(R 3) ² T _(R 4) ³ T _(R) ₅ ⁴ T _(R 6) ⁵ T _(R)

_(T) ⁰ T _(R)=₆ ⁰ T _(R T) ⁶ T _(R)

[0082] It leads to

_(T) ⁰ T _(R)=T_(D)

_(T) ⁰ T _(R=) ₆ ⁰ T _(R T) ⁶ T _(R)

₆ ⁰ T _(R)=_(T) ⁰ _(R T) ⁶ T _(R) ⁻¹ =T _(D T) ⁶ T _(R) ⁻¹

[0083] With the robotic inverse kinematics analysis, the angular displacement of the virtual ideal rigid robot can be obtained, and

θ_(V)={θ_(1V), θ_(2V),θ_(3V), θ_(4V), θ_(5V), θ_(6V)}

[0084] which is the absolute displacement with reference to the zero position of the joint. By using θ_(V) as the input of the joints of robot, if the robot has no error, then

_(T) ⁰ T _(fR)=_(T) ⁰ T _(R) =T _(D)

[0085] However, because of errors

_(T) ⁰ T _(fR)≠_(T) ⁰ T _(R)

[0086] At this moment, the robotic system will lose control since it cannot determine its end-effecter position. With the help of the shade system, the position of end-effecter frame of robot in the space of the shadow system can be calculated. The position of T_(T) within the space of the shadow system is _(T) ⁰T_(S)

_(T) ⁰ T _(S=) ₁ ⁰ T _(S 2) ¹ T _(S 3) ² T _(S 4) ³ T _(S 5) T _(S 6) ⁵ T _(S T) ⁶ T _(S)

[0087] and which is known since the 6-D shadow robotic system is considered almost ideal. T_(T) within the space of the shadow system is _(T) ⁰T_(fR), and it cannot be directly unknown. However, it can be obtained with the following process. It is known that  = T_(S)

[0088] Since

_(T) ⁰ T _(fR)=₆ ⁰ T _(fR) _(T) ⁶ T _(fR)

[0089] It leads =_(T)⁰T_(fR)⁻¹ = T_(S)⁻¹

[0090] By imagining the virtual robot to target the current misplaced position of the end-effecter frame, the solution of inverse kinematics is

θ_(f)={θ_(1f), θ_(2f), θ_(3f), θ_(4f), θ_(5f), θ_(6f)}

[0091] It is the actual angular displacements of the virtual robot. The difference caused by the errors is

θ_(V)−θ_(f)={θ_(1V)−θ_(1f), θ_(2V)−θ_(2f), θ_(3V)−θ_(3f), θ_(4V)−θ_(4f), θ_(5V)−θ_(5f), θ_(6V)−θ_(6f)}

[0092] that is,

δθ=θ_(V)−θ_(f)={δθ₁, δθ₂, δθ₃, δθ₄, δθ₅, δθ₆}

[0093] δθ is the feedback data needed for robot to achieve the desired position in the next step. To cover the difference and to achieve the desired position, the joint solution of inverse kinematics should have an extra displacement δθ. The total absolute displacement of the joint with reference to the zero position with the robot is

θ_(V)=θ_(V)+δθ

[0094] If the end-effecter position measured with the shadow system has not reached the desired position, then, the next step of the above process is needed. With the step-approaching method to repeat the above process step by step, the end-effecter of robot can finally achieve its desired position within limited steps.

[0095] Error Analysis of the Invention

[0096] Since the almost ideal shadow robotic measurement system is still not ideal, the error analysis is necessary to know the performance of the shadow system. It will particularly study the feasibility of the technology and demonstrate if the invention can achieve the expected positioning accuracy. With a working process design of the robotic system, the target position of end-effecter of the system can be written with a functional expression of

Y={P _(X) , P _(Y) , P _(Z), θ_(R), θ_(P), θ_(Y)}⁻¹ =F(Θ, Φ)

[0097] Here, P_(X), P_(Y), and P_(Z) are the origin. θ_(R), θ_(P), and θ_(Y) are rotational angles to the axes X, Y, and Z, which are also defined as Roll, Pitch, and Yaw in general. Θ={θ₁, θ₂, . . . , θ_(n)}, which are the n measured joint displacements of the n-D passive shadow system. Φ={100 ₁, φ₂, . . . , φ_(m)}, which are the m dimensional parameters with Denavit-Hartenberg (D-H) notation that are used to identify such a shadow system. With Taylor-series, the functional expression of the target can be further expressed as $Y = {{F\left( {\Theta^{nom},\Phi^{nom}} \right)} + {\sum\limits_{i = 1}^{m}\quad {\frac{\partial F}{\partial\varphi_{i}}{_{nom}{\left( {\varphi_{i} - \varphi_{1}^{nom}} \right) + {\sum\limits_{j = 1}^{n}\quad \frac{\partial F}{\partial\theta_{i}}}}}_{nom}\left( {\theta_{j} - \theta_{j}^{nom}} \right)}} + {\frac{1}{2!}\left\lbrack {\sum\limits_{i = 1}^{m}\quad {\frac{\partial^{2}F}{\partial\varphi_{i}^{2}}{_{nom}{\left( {\varphi_{1} - \varphi_{1}^{nom}} \right)^{2} + {\sum\limits_{j = 1}^{n}\quad \frac{\partial^{2}F}{\partial\theta_{i}^{2}}}}}_{nom}\left( {\theta_{j} - \theta_{j}^{nom}} \right)^{2}}} \right\rbrack} + {\sum\limits_{i > j}^{m}\quad {\frac{\partial^{2}F}{{\partial\varphi_{i}}{\partial\varphi_{i}}}{_{nom}{{\left( {\varphi_{1} - \varphi_{1}^{nom}} \right)\left( {\varphi_{j} - \varphi_{j}^{nom}} \right)} + {\sum\limits_{i > j}^{n}\quad \frac{\partial^{2}F}{{\partial\theta_{i}}{\partial\theta_{j}}}}}}_{nom}\left( {\theta_{1} - \theta_{1}^{nom}} \right)\left( {\theta_{j} - \theta_{j}^{nom}} \right)}} + \ldots}$

[0098] Here, Θ^(nom)={θ₁ ^(nom), θ₂ ^(nom), . . . , θ_(n) ^(nom)} are the nominal displacements measured with the shadow robotic measurement system. Φ^(nom)={φ₁ ^(nom), . . . , φ_(m) ^(nom)} are the nominal dimensions of the shadow system. For the small and independent variations of the nominal configurations, a linear approach of the Taylor series can be written as $Y \approx {{F\left( {\Theta^{nom},\Phi^{nom}} \right)} + \quad {\frac{\partial F}{\partial\Phi}{_{nom}{\left( {\Phi - \Phi^{nom}} \right) + \quad \frac{\partial F}{\partial\Theta}}}_{nom}\left( {\Theta - \Theta^{nom}} \right)}}$ It  leads  to ${\Delta \quad Y} \approx {{\frac{\partial F}{\partial\Phi}\Delta \quad \Phi} + {\frac{\partial F}{\partial\Theta}{\Delta\Theta}}}$

[0099] This is the analytic error of the shadow robotic measurement system. It relates the component variability ΔΦ in the D-H parameter space and measured angular displacement variability ΔΘ in the joint space of the shadow system to the output variation ΔY in Cartesian space, where, all higher order effects are neglected.

[0100] Since the worst performance of the shadow system can occur for any combination of minimum and maximum values of parameters ΔΦ and ΔΘ, to guarantee the implementation of the work of the proposed project, the worst-case error estimate will be employed. In the worst-case error estimate, the parameters ΔΘ and measured input variations ΔΘ are assumed to take the value ΔΦ_(min) or ΔΦ_(max) and ΔΘ_(min) or ΔΘ_(max) respectively. To each of the items, the maximum value of error is adopted, that is Δ  Y = Δ  F  Φ + Δ  F  Θ   = {Δ  P_(X), Δ  P_(Y), Δ  P_(Z), Δ  θ_(R), Δ  θ_(P), Δ  θ_(Y)}⁻¹

[0101] Here, {ΔP_(X), ΔP_(Y), ΔP_(Z)} are the positioning errors of origin; and {Δθ_(R), Δθ_(P), Δθ_(Y)} are the angular errors corresponding to Roll, Pitch, and Yaw. ΔF Φ is the maximum of the multiplied product of $\frac{\partial F}{\partial\Phi}$

[0102] and ΔΦ. ΔF Θ is the maximum of the multiplied product of $\frac{\partial F}{\partial\Theta}$

[0103] and ΔΘ. That is, $\begin{matrix} {{\Delta \quad F\quad \Phi} = {{Max}.\left\{ {{\frac{\partial F}{\partial\Phi}{\Delta\Phi}_{\min}},{\frac{\partial F}{\partial\Phi}{\Delta\Phi}_{\max}}} \right\}}} \\ {{\Delta \quad F\quad \Theta} = {{Max}.\left\{ {{\frac{\partial F}{\partial\Theta}\Delta \quad \Theta_{\min}},{\frac{\partial F}{\partial\Theta}\Delta \quad \Theta_{\max}}} \right\}}} \end{matrix}$

[0104] With the analysis of inverse kinematics, the joint error, Δθ, caused by ΔY can be determined, and

Δθ={Δθ₁, Δθ₂, Δθ₃, Δ₄, Δθ₅, Δθ₆}

[0105] which will be used to adjust the difference δθ to carry out more effective feedback control for the robotic system to more accurately achieve the desired position.

[0106] Example of Error Analysis with a Puma 560 Embodiment of Shadow Robotic Measurement System

[0107] To carry out the error analysis, a robotic structure of a 6-D Puma 560 robot is used here as an embodiment of the shadow robotic measurement system, except all the actuators of Puma 560 are replaced with optical rotary encoders which will be passively moved by the drawing force acting at the end-effecter connection-point (CP) generated by the robotic system (FIG. 5). The dimensions of the shadow robotic measurement system yielding the D-H parameters are a₂=1m, a₃=0.02m, a₆=0.1m, d₁=1m, d₃=0.15m, d₄=0.6m.

[0108] Derivation of the General Analytic Positioning Error of the System

[0109] The above maximum error analysis gives the basic idea about the limitation of the positioning accuracy of the robotic system. In the approaching process to achieve the desired position, with each of the steps, the results of analytic error analysis is needed to justify the solution of the inverse kinematics for the end-effecter of the robotic system to finally research its desired position. The processes of analytic error analysis are demonstrated below:

[0110] Determination of System Transformations:

[0111] The transformations of the Puma 560 shadow robotic measurement system can be determined as follows: $\begin{matrix} {{\,_{1}^{0}T} = \begin{pmatrix} c_{1} & {- s_{1}} & 0 & 0 \\ s_{1} & c_{1} & 0 & 0 \\ 0 & 0 & 1 & d_{1} \\ 0 & 0 & 0 & 1 \end{pmatrix}} & {{\,_{2}^{1}T} = \begin{pmatrix} c_{2} & {- s_{2}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- s_{2}} & {- c_{2}} & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}} & \quad \\ {{\,_{3}^{2}T} = \begin{pmatrix} c_{3} & {- s_{3}} & 0 & a_{2} \\ s_{3} & c_{3} & 0 & 0 \\ 0 & 0 & 1 & d_{3} \\ 0 & 0 & 0 & 1 \end{pmatrix}} & {{\,_{4}^{3}T} = \begin{pmatrix} c_{4} & {- s_{4}} & 0 & a_{3} \\ 0 & 0 & 1 & d_{4} \\ {- s_{4}} & {- c_{4}} & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}} & \quad \\ {{\,_{6}^{5}T} = \begin{pmatrix} c_{6} & {- s_{6}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {- s_{6}} & {- c_{6}} & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}} & {{\,_{7}^{6}T} = \begin{pmatrix} 1 & 0 & 0 & {- a_{6}} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}} & \quad \\ {{\,_{7}^{0}T} = \begin{pmatrix} r_{11} & r_{12} & r_{13} & P_{X} \\ r_{21} & r_{22} & r_{23} & P_{Y} \\ r_{31} & r_{32} & r_{33} & P_{Z} \\ 0 & 0 & 0 & 1 \end{pmatrix}} & \quad & \quad \end{matrix}$

r ₁₁ =c ₁ [c ₂₃(c ₄ c ₅ c ₆ −s ₄ s ₆)−s ₂₃ s ₅ c ₆ ]+s ₁(s ₄ c ₅ c ₆ +c ₄ s ₆)

r ₂₁ =s ₁ [c ₂₃ (c ₄ c ₅ c ₆ −s ₄ s ₆)−s ₂₃ s ₅ c ₆ ]−c ₁(s ₄ c ₅ c ₆ +c ₄ s ₆)

r ₃₁ =−s ₂₃(c ₄ c ₅ c ₆ −s ₄ s ₆)−c ₂₃ s ₅ c ₆

r ₁₂ =c ₁ [c ₂₃(−c ₄ c ₅ s ₆ −s ₄ c ₆)+s ₂₃ s ₅ s ₆ ]+s ₁(c ₄ c ₆ −s ₄ c ₅ s ₆)

r ₂₂ =s ₁ [c ₂₃(−c ₄ c ₅ s ₆ −s ₄ c ₆)+s ₂₃ s ₅ s ₆ ]−c ₁(c ₄ c ₆ −s ₄ c ₅ s ₆)

r ₃₂ =−s ₂₃(−c ₄ c ₅ s ₆ −s ₄ c ₆)+c ₂₃ s ₅ s ₆

r ₁₃ =−c ₁(c ₂₃ c ₄ s ₅ +s ₂₃ c ₅)−s ₁ s ₄ s ₅

r ₂₃ =−s ₁(c ₂₃ c ₄ s ₅ +s ₂₃ c ₅)+c ₁ s ₄ s ₅

r ₃₃ =s ₂₃ c ₄ s ₅ −c ₂₃ c ₅

P _(X) =c ₁(a ₂ c ₂ +a ₃ c ₂₃ −d ₄ s ₂₃)−d ₃ s ₁ —r ₁₁a₆

P _(Y) =s ₁(a ₂ c ₂ +a ₃ c ₂₃ −d ₄ s ₂₃)+d ₃ c ₁ −r ₂₁a₆

P _(Z) =−a ₃ s ₂₃ −a ₂ s ₂ −d ₄ c ₂₃ +d ¹⁻ r ₃₁a₆

[0112] Here,

c _(i)=cosθ_(i) and s _(i)=sinθ_(i)(i=1, 2, . . . , 6)

c ₂₃ =c ₂ c ₃ −s ₂ s ₃

s ₂₃ =c ₂ s ₃ +s ₂ c ₃

[0113] System Mathematical Model:

[0114] The mathematical model of the shadow system corresponding to the frame position of end-effecter CP is

Y=F(Θ, Φ)

[0115] which leads to

Y={P_(X), P_(Y), P_(Z), γ, β, α,}^(T)={P_(X), P_(Y), P_(Z), θ_(R), θ_(P), θ_(Y),}^(T)

[0116] Here,

θ_(R)=Atan2 (r ₃₂ , r ₃₃)

θ_(P)=Atan2 (−r ₃₁ , {square root}{square root over (r₁₁ ²+r₂₁ ²)})

θ_(Y)=Atan2 (r ₂₁ , r ₁₁)

[0117] and

Θ={θ₁, θ₂, θ₃, θ₄, θ₅, θ₆}

Φ={a₂, a₃, a₆, d₁, d₃, d₄)

[0118] System Error Analysis Model:

[0119] The analytical error model of the shadow system with the worst case can then be established as follows: ${\Delta \quad Y} \approx {{\frac{\partial F}{\partial\Phi}\quad {\Delta\Phi}} + {\frac{\partial F}{\partial\Theta}\quad {\Delta\Theta}}}$

[0120] which leads to $\begin{pmatrix} {\quad {\Delta \quad P_{X}}} \\ {\quad {\Delta \quad P_{Y}}} \\ {\Delta \quad P_{Z}} \\ {\Delta \quad \theta_{R}} \\ {\Delta \quad \theta_{P}} \\ {\Delta \quad \theta_{Y}} \end{pmatrix} = {\begin{pmatrix} {\quad P_{X}} \\ {\quad P_{Y}} \\ {\quad P_{Z}} \\ {\quad \theta_{R}} \\ {\quad \theta_{P}} \\ {\quad \theta_{Y}} \end{pmatrix}\begin{pmatrix} \begin{matrix} {\partial{/{\partial\theta_{1}}}} & {\partial{/{\partial\theta_{2}}}} & {\partial{/{\partial\theta_{3}}}} & {\partial{/{\partial\theta_{4}}}} & {\partial{/{\partial\theta_{5}}}} & {\partial{/{\partial\theta_{6}}}} \end{matrix} \\ \begin{matrix} {\partial{/{\partial a_{2}}}} & {\partial{/{\partial a_{3}}}} & {\partial{/{\partial a_{6}}}} & {\partial{/{\partial d_{1}}}} & {\partial{/{\partial d_{3}}}} & {\partial{/{\partial d_{4}}}} \end{matrix} \end{pmatrix}_{nom}\begin{pmatrix} {\Delta \quad \theta_{1}} \\ {\Delta \quad \theta_{2}} \\ {\Delta \quad \theta_{3}} \\ {\Delta \quad \theta_{4}} \\ {\Delta \quad \theta_{5}} \\ {\Delta \quad \theta_{6}} \\ {\Delta \quad a_{2}} \\ {\Delta \quad a_{3}} \\ {\Delta \quad a_{6}} \\ {\Delta \quad d_{1}} \\ {\Delta \quad d_{3}} \\ {\Delta \quad d_{4}} \end{pmatrix}}$

[0121] Here, Δθ₁, Δθ₂, Δθ₃, Δθ₄ , Δθ₅, and Δθ₆ are the angular displacement measurement errors of the shadow system in joint space, and Δa₂, Δa₃, Δa₆, Δd₁, Δd₃, and Δd₄ are the structural parameter measurement errors of the shadow system in the D-H parameter space. The linear dimensional errors, the angular displacement errors, and the absolute linear dimensional error of the frame origin can then be obtained as follows:

ΔD=±{ΔP _(X) , ΔP _(Y) , ΔP _(Z)}

Δθ=±{Δθ_(R), Δθ_(P), Δθ_(Y)}

|ΔD|={square root}{square root over (ΔP_(X) ²+ΔP_(Y) ²+ΔP_(Z) ²)}

[0122] Determine the Ranges of Measured Positioning Errors

[0123] To determine the ranges of measured positioning errors, the following simulation is designed with the change of each of joints of the shadow robotic measurement system. In this case, the joint measurement accuracy is yielding the resolution of the mated optical rotary encoder, which is ±0.28 arc-second. The D-H dimensional parameters can be determined with a specialized interferometer measurement platform with measurement error of ±1 nm. By using the above analytic positioning error model, the positioning error can be determined with the nominal joint displacements, joint angular measurement errors, and calibration errors of dimensional parameter. In this case, the arrangements and values of the simulation are given below, which lead to

Θ|_(nom={Δθ) ₁, Δθ₂, Δθ₃, Δθ₄, Δθ₅, Δθ₆}|_(nom)={θ_(1nom), θ_(2nom), θ_(3nom), θ_(4nom), θ_(5nom), θ_(6nom)}

Φ|_(nom)={a₂, a₃, a₆, d₁, d₃, d₄}={1, 0.02, 0.2, 1, 0.15, 0.6}(m)

ΔΘ={Δθ₁, Δθ₂, Δθ₃, Δθ₄, Δθ₅, Δθ₆}=±{0.28, 0.28, 0.28, 0.28, 0.28, 0.28} (arc-second)

ΔΦ={Δa₂, Δa₃, Δa₆, Δd₁, Δd₃, Δd₄}=±{1, 1, 1, 1, 1, 1} (nm)

[0124] To determine the range of measured positioning errors with the change of each of the joints, the algorithm of the simulation processes can be designed as follows:

[0125] Simulation Input Designation

[0126] Let θ_(inom)=(0°, 10°, 20° , . . . , 1080°), and fix the other θ_(j) at a random position in the robot joint space; here, θ_(jnom)ε[0, 360°] and (i=(1,2,3,4,5,6)∥j=(1,2,3,4,5,6)∥j≠i)

[0127] Obtain the Final Results of Positioning Errors With This Case

ΔD={ΔP_(X), ΔP_(Y), ΔP_(Z)}

Δθ={Δθ_(R), Δθ_(P), Δθ_(Y)}

|ΔD|={square root}{square root over (ΔP_(X) ²+ΔP_(Y) ²+ΔP_(Z) ²)}

[0128] Here, $\begin{matrix} \begin{matrix} {{\Delta \quad P_{X}} = \left\lbrack {{{{\partial P_{X}}/{\partial\theta_{1}}}{\Delta\theta}_{1}} + {{{\partial P_{X}}/{\partial\theta_{2}}}{\Delta\theta}_{2}} + {{{\partial P_{X}}/{\partial\theta_{3}}}{\Delta\theta}_{3}} +} \right.} \\ {{{{{\partial P_{X}}/{\partial\theta_{4}}}{\Delta\theta}_{4}} + {{{\partial P_{X}}/{\partial\theta_{5}}}{\Delta\theta}_{5}} + {{{\partial P_{X}}/{\partial\theta_{6}}}{\Delta\theta}_{6}} +}} \\ {{{{{\partial P_{X}}/{\partial a_{2}}}\Delta \quad a_{2}} + {{{\partial P_{X}}/{\partial a_{3}}}\Delta \quad a_{3}} + {{{\partial P_{X}}/{\partial a_{6}}}\Delta \quad a_{6}} +}} \\ \left. {{{{\partial P_{X}}/{\partial d_{3}}}\Delta \quad d_{3}} + {{{\partial P_{X}}/{\partial d_{4}}}\Delta \quad d_{4}}} \right\rbrack_{nom} \end{matrix} \\ \begin{matrix} {{\Delta \quad P_{Y}} = \left\lbrack {{{{\partial P_{Y}}/{\partial\theta_{1}}}{\Delta\theta}_{1}} + {{{\partial P_{Y}}/{\partial\theta_{2}}}{\Delta\theta}_{2}} + {{{\partial P_{Y}}/{\partial\theta_{3}}}{\Delta\theta}_{3}} +} \right.} \\ {{{{{\partial P_{Y}}/{\partial\theta_{4}}}{\Delta\theta}_{4}} + {{{\partial P_{Y}}/{\partial\theta_{5}}}{\Delta\theta}_{5}} + {{{\partial P_{Y}}/{\partial\theta_{6}}}{\Delta\theta}_{6}} +}} \\ {{{{{\partial P_{Y}}/{\partial a_{2}}}\Delta \quad a_{2}} + {{{\partial P_{Y}}/{\partial a_{3}}}\Delta \quad a_{3}} + {{{\partial P_{Y}}/{\partial a_{6}}}\Delta \quad a_{6}} +}} \\ \left. {{{{\partial P_{Y}}/{\partial d_{1}}}\Delta \quad d_{1}} + {{{\partial P_{Y}}/{\partial d_{3}}}\Delta \quad d_{3}} + {{{\partial P_{Y}}/{\partial d_{4}}}\Delta \quad d_{4}}} \right\rbrack_{{no}\quad m} \end{matrix} \\ \begin{matrix} {{\Delta \quad P_{Z}} = \left\lbrack {{{{\partial P_{Z}}/{\partial\theta_{1}}}{\Delta\theta}_{1}} + {{{\partial P_{Z}}/{\partial\theta_{2}}}{\Delta\theta}_{2}} + {{{\partial P_{Z}}/{\partial\theta_{3}}}{\Delta\theta}_{3}} +} \right.} \\ {{{{{\partial P_{Z}}/{\partial\theta_{4}}}{\Delta\theta}_{4}} + {{{\partial P_{Z}}/{\partial\theta_{5}}}{\Delta\theta}_{5}} + {{{\partial P_{Z}}/{\partial\theta_{6}}}{\Delta\theta}_{6}} +}} \\ {{{{{\partial P_{Z}}/{\partial a_{2}}}\Delta \quad a_{2}} + {{{\partial P_{Z}}/{\partial a_{3}}}\Delta \quad a_{3}} + {{{\partial P_{Z}}/{\partial a_{6}}}\Delta \quad a_{6}} +}} \\ \left. {{{{\partial P_{Z}}/{\partial d_{1}}}\Delta \quad d_{1}} + {{{\partial P_{Z}}/{\partial d_{3}}}\Delta \quad d_{3}} + {{{\partial P_{Z}}/{\partial d_{4}}}\Delta \quad d_{4}}} \right\rbrack_{nom} \end{matrix} \\ \begin{matrix} {{\Delta \quad \theta_{R}} = \left\lbrack {{{{\partial\theta_{R}}/{\partial\theta_{1}}}{\Delta\theta}_{1}} + {{{\partial\theta_{R}}/{\partial\theta_{2}}}{\Delta\theta}_{2}} + {{{\partial\theta_{R}}/{\partial\theta_{3}}}{\Delta\theta}_{3}} +} \right.} \\ {{{{{\partial\theta_{R}}/{\partial\theta_{4}}}{\Delta\theta}_{4}} + {{{\partial P_{Z}}/{\partial\theta_{5}}}{\Delta\theta}_{5}} + {{{\partial P_{Z}}/{\partial\theta_{6}}}{\Delta\theta}_{6}} +}} \\ {{{{{\partial\theta_{R}}/{\partial a_{2}}}\Delta \quad a_{2}} + {{{\partial\theta_{R}}/{\partial a_{3}}}\Delta \quad a_{3}} + {{{\partial\theta_{R}}/{\partial a_{6}}}\Delta \quad a_{6}} +}} \\ \left. {{{{\partial\theta_{R}}/{\partial d_{1}}}\Delta \quad d_{1}} + {{{\partial\theta_{R}}/{\partial d_{3}}}\Delta \quad d_{3}} + {{{\partial\theta_{R}}/{\partial d_{4}}}\Delta \quad d_{4}}} \right\rbrack_{nom} \end{matrix} \\ \begin{matrix} {{\Delta \quad \theta_{P}} = \left\lbrack {{{{\partial\theta_{P}}/{\partial\theta_{1}}}{\Delta\theta}_{1}} + {{{\partial\theta_{P}}/{\partial\theta_{2}}}{\Delta\theta}_{2}} + {{{\partial\theta_{P}}/{\partial\theta_{3}}}{\Delta\theta}_{3}} +} \right.} \\ {{{{{\partial\theta_{P}}/{\partial\theta_{4}}}{\Delta\theta}_{4}} + {{{\partial\theta_{P}}/{\partial\theta_{5}}}{\Delta\theta}_{5}} + {{{\partial\theta_{P}}/{\partial\theta_{6}}}{\Delta\theta}_{6}} +}} \\ {{{{{\partial\theta_{P}}/{\partial a_{2}}}\Delta \quad a_{2}} + {{{\partial\theta_{P}}/{\partial a_{3}}}\Delta \quad a_{3}} + {{{\partial\theta_{P}}/{\partial a_{6}}}\Delta \quad a_{6}} +}} \\ \left. {{{{\partial\theta_{P}}/{\partial d_{1}}}\Delta \quad d_{1}} + {{{\partial\theta_{P}}/{\partial d_{3}}}\Delta \quad d_{3}} + {{{\partial\theta_{P}}/{\partial d_{4}}}\Delta \quad d_{4}}} \right\rbrack_{nom} \end{matrix} \\ \begin{matrix} {{\Delta \quad \theta_{Y}} = \left\lbrack {{{{\partial\theta_{Y}}/{\partial\theta_{1}}}{\Delta\theta}_{1}} + {{{\partial\theta_{Y}}/{\partial\theta_{2}}}{\Delta\theta}_{2}} + {{{\partial\theta_{Y}}/{\partial\theta_{3}}}{\Delta\theta}_{3}}} \right.} \\ {{{{{\partial\theta_{Y}}/{\partial\theta_{4}}}{\Delta\theta}_{4}} + {{{\partial\theta_{Y}}/{\partial\theta_{5}}}{\Delta\theta}_{5}} + {{{\partial\theta_{Y}}/{\partial\theta_{6}}}{\Delta\theta}_{6}} +}} \\ {{{{{\partial\theta_{Y}}/{\partial a_{2}}}\Delta \quad a_{2}} + {{{\partial\theta_{Y}}/{\partial a_{3}}}\Delta \quad a_{3}} + {{{\partial\theta_{Y}}/{\partial a_{6}}}\Delta \quad a_{6}} +}} \\ \left. {{{{\partial\theta_{Y}}/{\partial d_{1}}}\Delta \quad d_{1}} + {{{\partial\theta_{Y}}/{\partial d_{3}}}\Delta \quad d_{3}} + {{{\partial\theta_{Y}}/{\partial d_{4}}}\Delta \quad d_{4}}} \right\rbrack_{nom} \end{matrix} \end{matrix}$

[0129] The above solution is only true when |cosθ_(P)|>0. If |cosθ_(P)|=0, the solution will degenerate. In such cases, only the sum or the difference of θ_(R) and θ_(Y) can be computed.

[0130] Draw Curves of Simulated Positioning Errors

[0131] The corresponding positioning errors with the shadow robotic measurement system can then be described through FIG. 6 to FIG. 11.

[0132] The simulations show that within most of its working envelope the measured positioning errors with the Puma 560 shadow system is smaller than the estimated maximum errors of an extreme 6-D system which is

ERR={ΔD, Δθ}=±{7.21 nm, 1.68 arc-second}

[0133] With the highly correlated space, the simulation results also show that there are still lots of space to realize the expected goals.

[0134] Although the results of error analysis are conducted by the use of a common optical rotary encoder on the market, such results have already conducted the breakthrough for the development of robotics and ultra-precision engineering. As above-mentioned, if the specialized ultra-precision optical rotary encoder can be used to build the shadow system, more precise positioning accuracy for either rigid body robotic system or a LDRS is expected. As known, to date, the robotics technology is still in its developing stage for pursuing the positioning accuracy in micro-accuracy level. In this situation, the advantages of the technology of ultra-precision robotic system developed with this research are obvious. 

Having presented my invention, I claim:
 1. An ultra-precision robotic system comprises a robot, a shadow robotic measurement system, and a robotic control; whereby the said shadow robotic measurement system mating with said robot to monitor the real-time position of the end-effecter of said robot and to collect the monitored real-time position data of the end-effecter of said robot which include the frame origin position and frame rotation of pitch, roll and yaw.
 2. An ultra-precision robotic system comprises a robot, a shadow robotic measurement system, and a robotic control; whereby the said robotic control uses the real-time data of the end-effecter position of the said robot that are collected by the said shadow robotic measurement system to control the end-effecter of the said robot to achieve the designated position.
 3. An ultra-precision robotic system comprises a robot, a shadow robotic measurement system, and a robotic control; whereby the said robot is either a robot under rigid body guidance or a robot with large deformation analysis (LDRS, i.e. a semi-flexible or a flexible robot).
 4. The ultra-precision robotic system according to claim 1 wherein said robotic control is integrated with the control system of the said robot to control the positioning process with reference of the real-time data of the end-effecter position of the said robot collected by the said shadow robotic measurement system.
 5. The ultra-precision robotic system according to claim 1 wherein said robotic control uses the stepping technique to converge the end-effecter of the said robot at the designated position with reference of the real-time data of the end-effecter position of the said robot collected by the said shadow robotic measurement system.
 6. The shadow robotic measurement system according to claim 2 consists of linkages, kinematic joints, and sensors that are mated with the kinematic joints to form a passive servo-measurement system with proper degree-of-freedom to monitor the position of the end-effecter of the said robot with its own end-effecter through the connection-point.
 7. The ultra-precision robotic system according to claim 1 wherein said shadow robotic measurement system is with proper degree-of-freedom to monitor any chosen positions on the said robot with its own end-effecter through the connection-point.
 8. The said shadow robotic measurement system according to claim 6 wherein the said kinematic joints can be either revolute joints or prismatic joints.
 9. The said shadow robotic measurement system according to claim 8 wherein the said revolute joints are mated with proper rotary measurement sensors to monitor and record the angular displacement of the said revolute joints.
 10. The said shadow robotic measurement system according to claim 8 wherein the said prismatic joints are mated with proper linear measurement sensors to monitor and record the linear displacement of the said prismatic joints. 